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The main objective of this study is to introduce a new class of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>q</mi> </mrow> </semantics> </math> </inline-formula>-point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs.